This document summarizes a paper that examines the performance of interior point methods for solving linear optimization problems. It presents a refined version of the Klee-Minty cubes problem that forces the central path of interior point methods to visit all 2n vertices of an n-dimensional cube. The key results are: 1) For this problem, the central path must make at least 2n-2 turns before converging to the optimal solution, providing a lower bound on the iteration complexity of interior point methods. 2) The upper bound on iteration complexity for this problem is O(2n*n5/2), nearly matching the lower bound and showing interior point methods can perform close to worst-case on this

1. Math. Program., Ser. A (2008) 113:1–14
DOI 10.1007/s10107-006-0044-x
F U L L L E N G T H PA P E R
How good are interior point methods? Klee–Minty
cubes tighten iteration-complexity bounds
Antoine Deza · Eissa Nematollahi ·
Tamás Terlaky
Received: 7 January 2005 / Revised: 16 August 2006 / Published online: 21 October 2006
© Springer-Verlag 2006
Abstract By refining a variant of the Klee–Minty example that forces the cen-
tral path to visit all the vertices of the Klee–Minty n-cube, we exhibit a nearly
worst-case example for path-following interior point methods. Namely, while
5
the theoretical iteration-complexity upper bound is O(2n n 2 ), we prove that
solving this n-dimensional linear optimization problem requires at least 2n − 1
iterations.
Keywords Linear programming · Interior point method · Worst-case
iteration-complexity
Mathematics Subject Classification (2000) Primary 90C05;
Secondary 90C51 · 90C27 · 52B12
1 Introduction
While the simplex method, introduced by Dantzig [1], works very well in practice
for linear optimization problems, in 1972 Klee and Minty [6] gave an exam-
ple for which the simplex method takes an exponential number of iterations.
Dedicated to Professor Emil Klafszky on the occasion of his 70th birthday.
A. Deza · E. Nematollahi · T. Terlaky (B)
Advanced Optimization Laboratory, Department of Computing and Software,
McMaster University, Hamilton, ON, Canada
e-mail: terlaky@mcmaster.ca
A. Deza
e-mail: deza@mcmaster.ca
E. Nematollahi
e-mail: nematoe@mcmaster.ca

2. 2 A. Deza et al.
More precisely, they considered a maximization problem over an n-dimensional
“squashed” cube and proved that a variant of the simplex method visits all its 2n
vertices. Thus, the time complexity is not polynomial for the worst case, as 2n −1
iterations are necessary for this n-dimensional linear optimization problem. The
pivot rule used in the Klee–Minty example was the most negative reduced cost
but variants of the Klee–Minty n-cube allow to prove exponential running time
for most pivot rules; see [11] and the references therein. The Klee–Minty worst-
case example partially stimulated the search for a polynomial algorithm and,
in 1979, Khachiyan’s [5] ellipsoid method proved that linear programming is
indeed polynomially solvable. In 1984, Karmarkar [4] proposed a more effi-
cient polynomial algorithm that sparked the research on polynomial interior
point methods. In short, while the simplex method goes along the edges of the
polyhedron corresponding to the feasible region, interior point methods pass
through the interior of this polyhedron. Starting at the analytic center, most inte-
rior point methods follow the so-called central path and converge to the analytic
center of the optimal face; see e.g. [7, 9, 10, 14, 15]. In 2004, Deza et al. [2] showed
that, by carefully adding an exponential number of redundant constraints to the
Klee–Minty n-cube, the central path can be severely distorted. Specifically, they
provided an example for which path-following interior point methods have to
take 2n − 2 sharp turns as the central path passes within an arbitrarily small
neighborhood of the corresponding vertices of the Klee–Minty cube before
converging to the optimal solution. This example yields a theoretical lower
bound for the number of iterations needed for path-following interior point
methods: the number of iterations is at least the number of sharp turns; that is,
n ).
the iteration-complexity lower bound is (2√ On the other hand, the theoret-
ical iteration-complexity upper bound is O( NL) where N and L respectively
denote the number of constraints and the bit-length of the input-data. The
iteration-complexity upper bound for the highly redundant Klee–Minty n-cube
of [2] is O(23n nL) = O(29n n4 ), as N = O(26n n2 ) and L = O(26n n3 ) for this
example. Therefore, these 2n − 1 sharp turns yield an ( 6 lnNN ) iteration-com-
2
plexity lower bound. In this paper we show that a refined problem with the same
(2n ) iteration-complexity lower bound exhibits a nearly worst-case iteration-
5
complexity as the complexity upper bound is O(2n n 2 ). In other words, this new
example, with N = O(22n n3 ), essentially closes the iteration-complexity gap
√
with an ( lnNN ) lower bound and an O( N ln N) upper bound.
3
2 Notations and the main results
We consider the following Klee–Minty variant where ε is a small positive factor
by which the unit cube [0, 1]n is squashed.
min xn ,
subject to 0 ≤ x1 ≤ 1,
ε xk−1 ≤ xk ≤ 1 − ε xk−1 for k = 2, . . . , n.

3. How good are interior point methods? 3
The above minimization problem has 2n constraints, n variables and the feasible
region is an n-dimensional cube denoted by C. Some variants of the simplex
method take 2n − 1 iterations to solve this problem as they visit all the ver-
tices ordered by the decreasing value of the last coordinate xn starting from
v{n} = (0, . . . , 0, 1) till the optimal value x∗ = 0 is reached at the origin v∅ .
n
While adding a set h of redundant inequalities does not change the feasible
region, the analytic center χ h and the central path are affected by the addi-
tion of redundant constraints. We consider redundant inequalities induced by
hyperplanes parallel to the n facets of C containing the origin. The constraint
parallel to the facet H1 : x1 = 0 is added h1 times at a distance d1 and the
constraint parallel to the facet Hk : xk = εxk−1 is added hk times at a distance
dk for k = 2, . . . , n. The set h is denoted by the integer-vector h = (h1 , . . . , hn ),
d = (d1 , . . . , dn ), and the redundant linear optimization problem is defined by
min xn
subject to 0 ≤ x1 ≤ 1
ε xk−1 ≤ xk ≤ 1 − ε xk−1 for k = 2, . . . , n
0 ≤ d1 + x1 repeated h1 times
ε x1 ≤ d2 + x2 repeated h2 times
.
. .
.
. .
ε xn−1 ≤ dn + xn repeated hn times.
By analogy with the unit cube [0, 1]n , we denote the vertices of the Klee–Minty
cube C by using a subset S of {1, . . . , n}, see Fig. 1. For S ⊂ {1, . . . , n}, a vertex
vS of C is defined by
1, if 1 ∈ S
vS =
1 0, otherwise
1 − εvS , if k ∈ S
vS = k−1 k = 2, . . . , n.
k εvS ,
k−1 otherwise
The δ-neighborhood Nδ (vS ) of a vertex vS is defined, with the convention x0 = 0,
by
1 − xk − εxk−1 ≤ εk−1 δ, if k ∈ S
Nδ (vS ) = x ∈ C : k = 1, . . . , n .
xk − εxk−1 ≤ εk−1 δ, otherwise
In this paper we focus on the following problem Cn defined by
δ
ε= n
2(n+1) ,
d= n(2n+4 , . . . , 2n−k+5 , . . . , 25 ),
22n+8 (n+1)n 2n+7 (n+1) 22n+8 (n+1)n+k−1
h= δnn−1
− δ ,..., δnn+k−2

4. 4 A. Deza et al.
Fig. 1 The δ-neighborhoods
v {2}
of the four vertices of the
Klee–Minty 2-cube
v {1,2}
v {1}
v∅
n+k+6 (n+1)2k−1 2n+6 (n+1)2n−1
−2 δn2k−2
, . . . , 32 δn2n−2
,
where 0 < δ ≤ 1
4(n+1) .
Note that we have: ε + δ < 1 ; that is, the δ-neighborhoods of the 2n vertices
2
are non-overlapping, and that h is, up to a floor operation, linearly dependent
on δ −1 . Proposition 2.1 states that, for Cn , the central path takes at least 2n − 2
δ
turns before converging to the origin as it passes through the δ-neighborhood
of all the 2n vertices of the Klee-Minty n-cube; see Sect. 3.2 for the proof. Note
that the proof given in Sect. 3.2 yields a slightly stronger result than Propo-
sition 2.1: In addition to pass through the δ-neighborhood of all the vertices,
the central path is bent along the edge-path followed by the simplex method.
We set δ = 4(n+1) in Propositions 2.3 and 2.4 in order to exhibit the sharpest
1
bounds. The corresponding linear optimization problem Cn 1/4(n+1) depends only
on the dimension n.
Proposition 2.1 The central path P of Cn intersects the δ-neighborhood of each
δ
vertex of the n-cube.
Since the number of iterations required by path-following interior point meth-
ods is at least the number of sharp turns, Proposition 2.1 yields a theoretical
lower bound for the iteration-complexity for solving this n-dimensional linear
optimization problem.
Corollary 2.2 For Cn , the iteration-complexity lower bound of path-following
δ
interior point methods is (2n ).
Since the theoretical iteration-complexity upper bound for path-following
√
interior point methods is O NL , where N and L respectively denote the
number of constraints and the bit-length of the input-data, we have:
Proposition 2.3 For Cn
1/4(n+1) , the iteration-complexity upper bound of path-
3 11
following interior point methods is O(2n n 2 L); that is, O(23n n 2 ).

5. How good are interior point methods? 5
n+k
n n
Proof We have N = 2n + k=1 hk = 2n + k=1 n
2 22n+10 n+1
n − 2n+k+8
2k n+k
n
n+1
n and, since k=1
n+1
n ≤ ne2 , we have N = O(22n n3 ) and L ≤
N ln d1 = O(22n n4 ).
Noticing that the only two vertices with last coordinates smaller than or equal
{1}
to εn−1 are v∅ and v{1} , with v∅ = 0 and vn = εn−1 , the stopping criterion can be
n
replaced by: stopping duality gap smaller than εn with the corresponding central
path parameter at the stopping point being µ∗ = ε . Additionally, one can check
n
N
that by setting the central path parameter to µ0 = 1, we obtain a starting point
which belongs to the interior of the δ-neighborhood of the highest vertex v{n} ,
see Sect. 3.3 for a detailed proof. In other words, a path-following algorithm
using a standard -precision as stopping criterion can stop when the duality
gap is smaller than εn as the optimal vertex is identified, see [9]. The corre-
√
sponding iteration-complexity bound O( N log N ) yields, for our construction,
√ Nµ0 √
a precision-independent iteration-complexity O( N ln Nµ∗ ) = O( Nn) and
Proposition 2.3 can therefore be strengthened to:
Proposition 2.4 For Cn
1/4(n+1) , the iteration-complexity upper bound of path-
5
following interior point methods is O(2n n 2 ).
Remark 2.5
(i) For Cn
1/4(n+1) , by Corollary 2.2 and Proposition 2.4, the order of the iter-
ation-complexity of path-following interior point methods is between 2n
5 √
and 2n n 2 or, equivalently, between lnNN and N ln N.
3
(ii) The k-th coordinate of the vector d corresponds to the scalar d defined
in [2] for dimension n − k + 3.
(iii) Other settings for d and h ensuring that the central path visits all the
vertices of the Klee–Minty n-cube are possible. For example, d can be set
to (1.1, 22) in dimension 2.
(iv) Our results apply to path-following interior point methods but not to
other interior point methods such as Karmarkar’s original projective
algorithm [4].
Remark 2.6
(i) Megiddo and Schub [8] proved, for affine scaling trajectories, a result
with a similar flavor as our result for the central path, and noted that
their approach does not extend to projective scaling. They considered
the non-redundant Klee–Minty √ cube.
(ii) Todd and Ye [12] gave an ( 3 N) iteration-complexity lower bound
between two updates of the central path parameter µ.
(iii) Vavasis and Ye [13] provided an O(N 2 ) upper bound for the number of
approximately straight segments of the central path.

6. 6 A. Deza et al.
(iv) A referee pointed out that a knapsack problem with proper objective
function yields an n-dimensional example with n + 1 constraints and n
sharp turns.
(v) Deza et al. [3] provided a non-redundant construction with N constraints
and N − 4 sharp turns.
3 Proofs of Proposition 2.1 and Proposition 2.4
3.1 Preliminary lemmas
˜
Lemma 3.1 With b= 4 (1, . . . , 1), ε= 2(n+1) , d=n(2n+4 , . . . , 2n−k+5 , . . . , 25 ), h =
n
δ
22n+8 (n+1)n 2n+7 (n+1) 2n+8 (n+1)n+k−1 n+k+6 (n+1)2k−1 2n+6 (n+1)2n−1
δnn−1
− δ , . . . , 2 δnn+k−2 −2 δn2k−2
, . . . , 32 δn2n−2
and
⎛ −ε ⎞
1
d1 +1 d2 0 0 ... 0 0
⎜ −1 −ε2 ⎟
⎜ 2ε
d2 +1 0 ... 0 0 ⎟
⎜ d1 d3 ⎟
⎜ . . .. .. . . . ⎟
⎜ .
. .
. . . .
. .
. .
. ⎟
⎜ ⎟
⎜ −1 2εk−1 −εk ⎟
A=⎜ 0 0 dk +1 0 0 ⎟,
⎜ d1 dk+1
⎟
⎜ .
. .
. .
. .
. .. .. ⎟
⎜ . . . . . . 0 ⎟
⎜ ⎟
⎜ −1
0 0 0 ... 2εn−2 −εn−1 ⎟
⎝ d1 dn−1 +1 dn ⎠
−1 2εn−1
d1 0 0 0 ... 0 dn +1
˜
we have Ah ≥ 3b
2 .
˜
Proof As ε = 2(n+1) and d = n(2n+4 , . . . , 2n−k+5 , . . . , 25 ), h can be rewritten as
n
˜ ˜
h = 4 d1 ( ε4n − 1 ), . . . , εdk ( ε4n − ε1k ), . . . , εdn ε3n and Ah ≥ 3b can be rewritten
δ ε k−1 n−1 2
as
4 d1 4 1 4 4 1 6
− − − 2 ≥
δ d1 + 1 εn ε δ εn ε δ
4 4 1 4 2dk 4 1 4 4 1 6
− − + − − − k+1 ≥ for k = 2, . . . , n−1
δ εn ε δ dk + 1 εn εk δ εn ε δ
4 4 1 4 2dn 3 6
− − + ≥ ,
δ εn ε δ dn + 1 εn δ
which is equivalent to
1 1 3 4 1 3
− − d1 ≥ n − 2 + ,
ε2 ε 2 ε ε 2
1 2 1 3 8 1 1 3
− k+ − dk ≥ n − k+1 − + for k = 2, . . . , n − 1,
εk+1 ε ε 2 ε ε ε 2

7. How good are interior point methods? 7
2 1 3 4 1 3
+ − dn ≥ n − + .
ε n ε 2 ε ε 2
As ε12 − 1 − 3 ≥ 1 ,
ε 2 2
1
ε − 3
2 ≥ 0, 1
ε2
− 3
2 ≥ 0 and 1
εk+1
+ 1
ε − 3
2 ≥ 0, the above
system is implied by
1 4
d1 ≥ n ,
2 ε
1 2 8
− k dk ≥ n for k = 2, . . . , n − 1,
εk+1 ε ε
2 4
d ≥ n,
n n
ε ε
n−k
as εk+1 − ε2k =
1 2
nεk
and 1
εn−k
= 2n−k 1 + 1
n ≤ 2n−k+2 , the above system is
implied by
d1 ≥ 2n+5 ,
dk ≥ n2n−k+4 for k = 2, . . . , n − 1,
dn ≥ 2,
which is true since d = n(2n+4 , . . . , 2n−k+5 , . . . , 25 ).
˜
Corollary 3.2 With the same assumptions as in Lemma 3.1 and h = h , we have
Ah ≥ b.
˜
Proof Since 0 ≤ hk − hk < 1 and dk = n2n−k+5 , we have:
˜
h1 − h1 ˜
(h2 − h2 )ε 2
− ≤ ,
d1 + 1 d2 δ
˜
h1 − h1 ˜
2(hk − hk )εk−1 ˜
(hk+1 − hk+1 )εk 2
− + − ≤ for k = 2, . . . , n − 1,
d1 dk + 1 dk+1 δ
˜
h1 − h1 ˜
2(hn − hn )εn−1 2
− + ≤ ,
d1 dn + 1 δ
˜ ˜
thus, A(h − h) ≤ b , which implies, since Ah ≥ 3b
by Lemma 3.1, that Ah ≥ b.
2 2
˜
Corollary 3.3 With the same assumptions as in Lemma 3.1 and h = h , we have:
hk εk−1 hk+1 εk
dk +1 ≥ dk+1 + 4
δ for k = 1, . . . , n − 1.
Proof For k = 1, . . . , n − 1, one can easily check that the first k inequalities of
hk εk−1 hk+1 εk
Ah ≥ b imply dk +1 ≥ dk+1 + 4.
δ

8. 8 A. Deza et al.
The analytic center χ n = (ξ1 , . . . , ξn ) of Cn is the unique solution to the
n n
δ
problem consisting of maximizing the product of the slack variables:
s1 = x1
sk = xk − εxk−1 for k = 2, . . . , n
¯
s1 = 1 − x1
¯
sk = 1 − εxk−1 − xk for k = 2, . . . , n
˜
s1 = d1 + s1 repeated h1 times
.
. .
.
. .
˜
sn = dn + sn repeated hn times.
Equivalently, χ n is the solution of the following maximization problem:
n
max ¯ ˜
ln sk + ln sk + hk ln sk ,
x
k=1
i.e., with the convention x0 = 0,
n
max ln(xk − εxk−1 ) + ln(1 − εxk−1 − xk ) + hk ln(dk + xk − εxk−1 ) .
x
k=1
The optimality conditions (the gradient is equal to zero at optimality) for this
concave maximization problem give:
⎧ hk+1 ε
ε ε hk
⎪ σ1n −
⎨ σk+1
n − 1
σk
¯n
− σk+1
¯n
+ σk
˜n
−
σk+1
˜n
= 0 for k = 1, . . . , n − 1,
k
1
n − σn + σn
1 hn
=0 (1)
⎪
⎩ σn ¯n ˜n
σk > 0, σk > 0, σk
n ¯n ˜n > 0 for k = 1, . . . , n,
where
n n
σ1 = ξ1
n n n
σk = ξk − εξk−1 for k = 2, . . . , n,
n n
σ1 = 1 − ξ1
¯
¯n n n
σk = 1 − εξk−1 − ξk for k = 2, . . . , n,
˜n
σk = dk + σkn
for k = 1, . . . , n.
The following lemma states that, for Cn , the analytic center χ n belongs to the
δ
neighborhood of the vertex v{n} = (0, . . . , 0, 1).
Lemma 3.4 For Cn , we have: χ n ∈ Nδ (v{n} ).
δ

9. How good are interior point methods? 9
Proof Adding the nth equation of (1) multiplied by −εn−1 to the jth equation
of (1) multiplied by εj−1 for j = k, . . . , n − 1, we have, for k = 1, . . . , n − 1,
n−2
εk−1 εk−1 2εn−1 εi hk εk−1 2hn εn−1
n − σn − σn − 2
σk ¯k n +
σi+1
¯ σk
˜ n −
σn
˜n
= 0,
n i=k
implying:
n−2
2hn εn−1 hk εk−1 εk−1 εk−1 2εn−1 εi εk−1
− = n − n + σn + 2 ≤ n ,
σn
˜ n σk
˜ n σk σk
¯ n σi+1
¯n σk
i=k
h1 hk εk−1
which implies, since σn ≤ dn + 1, σk ≥ dk and
˜n ˜n d1 ≥ dk by Corollary 3.3,
2hn εn−1 h1 εk−1
− ≤ n ,
dn + 1 d1 σk
implying, since 2hn ε − h1 ≥
n−1
dn +1 d
1
δ by Corollary 3.2, σk ≤ εk−1 δ for k = 1, . . . , n−1.
n
1
hn εn−1 εn−1
The n-th equation of (1) implies: σn
˜n
≤ σn
¯n
; that is, since σn < dn + 1 and
˜n
hn εn−1 hn ε n−1
εn−1
dn +1 ≥ 1
δ by Corollary 3.2, we have: 1 δ ≤ dn +1 ≤ σn
¯n
, implying: σn ≤ εn−1 δ.
¯n
The central path P of Cn can be defined as the set of analytic centers χ n (α) =
δ
(xn , . . . , xn , α) of the intersection of the hyperplane Hα : xn = α with the
1 n−1
feasible region of Cn where 0 < α ≤ ξn , see [9]. These intersections (α) are
δ
n
called the level sets and χ n (α) is the solution of the following system:
ε ε hk hk+1 ε
1
sn
− sn
− 1
¯k
sn
− ¯k+1 + sn − sn
sn ˜k ˜k+1 =0 for k = 1, . . . , n − 1
k k+1 (2)
k ¯k ˜k
sn > 0, sn > 0, sn >0 for k = 1, . . . , n − 1,
where
sn
1 = xn
1
sn
k = xn − εxn
k k−1 for k = 2, . . . , n − 1,
sn
n = α − εxn−1
¯1
sn = 1 − xn
1
¯k
sn = 1 − εxn − xn
k−1 k for k = 2, . . . , n − 1,
¯n
sn = 1 − α − εxn
n−1
˜k
sn = dk + sn
k for k = 1, . . . , n.
¯ ˆk
Lemma 3.5 For Cn , Cδ = {x ∈ C : sk ≥ εk−1 δ, sk ≥ εk−1 δ} and Cδ = {x ∈ C :
k
δ
¯ ˆk
sk−1 ≤ εk−2 δ, sk−2 ≤ εk−3 δ, . . . , s1 ≤ δ}, we have: Cδ ∩ P ⊆ Cδ for k = 2, . . . , n.
k

10. 10 A. Deza et al.
Proof Let x ∈ Cδ ∩ P. Adding the (k − 1)th equation of (2) multiplied by −εk−2
k
to the ith equation of (1) multiplied by εi−1 for i = j . . . , k − 2, we have, for
k = 2, . . . , n − 1,
2hk−1 εk−2 hj εj−1 hk εk−1 εj−1 εk−1 εk−1
− + + + n + n + n
˜k−1
sn ˜j
sn ˜k
sn sj sk ¯
sk
⎛ ⎞
k−3
2εk−2 εj−1 εi ⎠
−⎝ n + n +2 = 0,
sk−1 ¯
sj ¯ n
si+1
i=j
˜k−1 ˜j ˜k
which implies, since sn < dk−1 + 1, sn > dj , sn > dk and sn ≥ εk−1 δ and
k
¯k
sn ≥ εk−1 δ as x ∈ Cδ ,
k
2hk−1 εk−2 hj εj−1 hk εk−1 εj−1 2
− − ≤ n + ,
dk−1 + 1 dj dk sj δ
h1 hj εj−1
implying, since d1 ≥ dj by Corollary 3.3,
h1 2hk−1 εk−2 hk εk−1 εj−1 2
− + − ≤ n + ,
d1 dk−1 + 1 dk sj δ
2h εk−2
ε k−1
that is, as 3 ≤ − h1 + dk−1 +1 − hkd by Corollary 3.2: sn ≤ εj−1 δ. Considering
δ d1 j
k−1 k
the (k − 1)th equation of (2), we have
hk−1 εk−2 hk εk−1 εk−2 εk−1 εk−1 εk−2
− = n + n + n − n ,
˜ n
sk−1 ˜
skn ¯
sk−1 sk ¯
sk sk−1
˜k−1 ˜k ¯k
which implies, since sn < dk−1 + 1, sn > dk and sn ≥ εk−1 δ and sn ≥ εk−1 δ as
k
x ∈ Cδ ,
k
hk−1 εk−2 hk εk−1 εk−2 2
− ≤ n + ,
dk−1 + 1 dk ¯
sk−1 δ
hk−1 εk−2 hk εk−1
which implies, since 3 ≤
δ dk−1 +1 − dk ¯k−1
by Corollary 3.3, that sn ≤ εk−2 δ
ˆ k.
and, therefore, x ∈ C δ
3.2 Proof of Proposition 2.1
For k = 2, . . . , n, while Cδ , defined in Lemma 3.5, can be seen as the central part
k
k ¯
of the cube C, the sets Tδ = {x ∈ C : sk ≤ εk−1 δ} and Bk = {x ∈ C : sk ≤ εk−1 δ},
δ

11. How good are interior point methods? 11
Fig. 2 The set Pδ for the Klee–Minty 3-cube
3
T0
ˆ3
C0
2
B0 ˆ2
C0 2
T0
3
B0
A2
0 A3
0
Fig. 3 The sets A2 and A3 for the Klee–Minty 3-cube
0 0
can be seen, respectively, as the top and bottom part of C. Clearly, we have
k k ˆk
C = Tδ ∪ Cδ ∪ Bk for each k = 2, . . . , n. Using the set Cδ defined in Lemma 3.5,
δ
k ˆk
we consider the set Ak = Tδ ∪ Cδ ∪ Bk for k = 2 . . . , n, and, for 0 < δ ≤ 4(n+1) ,
1
δ δ
we show that the set Pδ = n Ak , see Fig. 2, contains the central path P. By
k=2 δ
Lemma 3.4, the starting point χ n of P belongs to Nδ (v{n} ). Since P ⊂ C and
C = n (Tδ ∪ Cδ ∪ Bk ), we have:
k=2
k k
δ
n n
P=C∩P= Tδ ∪ Cδ ∪ Bk ∩ P =
k k
δ Tδ ∪ Cδ ∩ P ∪ Bk ∩ P,
k k
δ
k=2 k=2
that is, by Lemma 3.5,
n n
P⊆ k ˆk
Tδ ∪ Cδ ∪ Bk = Ak = Pδ
δ δ
k=2 k=2
k ˆk
Remark that the sets Cδ , Cδ , Tδ , Bk and Ak can be defined for δ = 0, see Fig. 3,
k
δ δ
and that the corresponding set P0 = n Ak is precisely the path followed
k=2 0
by the simplex method on the original Klee-Minty problem as it pivots along
the edges of C. The set Pδ is a δ-sized (cross section) tube along the path P0 .
See Fig. 4 illustrating how P0 starts at v{n} , decreases with respect to the last
coordinate xn and ends at v∅ .

12. 12 A. Deza et al.
Fig. 4 The path P0 followed
v {3}
by the simplex method for the
Klee–Minty 3-cube
v {1,3} v {2,3}
v {1,2,3}
v {2}
v {1,2}
∅
v v {1}
3.3 Proof of Proposition 2.4
¯
We consider the point x of the central path which lies on the boundary of the
δ-neighborhood of the highest vertex v{n} . This point is defined by: s1 = δ, sk ≤
εk−1 δ for k = 2, . . . , n − 1 and s2n ≤ εn δ. Note that the notation sk for the cen-
tral path (perturbed complementarity) conditions, yk sk = µ for k = 1, . . . , pn ,
is consistent with the slacks introduced after Corollary 3.3 with sn+k = sk for ¯
˜
k = 1, . . . , n and spi +k = sk for k = 1, . . . , hi+1 Ê and i = 0, . . . , n − 1. Let µ
¯
¯
denote the central path parameter corresponding to x. In the following, we
prove that µ ≤ εn−1 δ which implies that any point of the central path with
¯
corresponding parameter µ ≥ µ belong to the interior of the δ-neighborhood
¯
of the highest vertex v{n} . In particular, it implies that by setting the central path
parameter to µ0 = 1, we obtain a starting point which belongs to the interior of
the δ-neighborhood of the vertex v{n} .
3.3.1 Estimation of the central path parameter µ
¯
The formulation of the dual problem of Cn is:
δ
2n n pk
max z=− yk − dk yi
k=n+1 k=1 i=pk−1 +1
subject to yk − εyk+1 − yn+k − εyn+k+1
pk pk+1
+ yi − ε yi = 0 for k = 1, . . . , n − 1
i=pk−1 +1 i=pk +1
pn
yn − y2n + yi = 1
i=pn−1 +1
yk ≥ 0 for k = 1, . . . , pn ,
where p0 = 2n and pk = 2n + h1 + · · · + hk for k = 1, . . . , n.

13. How good are interior point methods? 13
For k = 1, . . . , n, multiplying by εk−1 the kth equation of the above dual
constraints and summing then up, we have:
2n+h1
y1 − yn+1 − 2 εyn+2 + ε2 yn+3 + · · · + εn−1 y2n + yi = εn−1
i=2n+1
which implies
2n+h1
n−1
2ε y2n ≤ y1 + yi
i=2n+1
µ¯
implying, since for i = 2n + 1, . . . , 2n + h1 , d1 ≤ si yields yi ≤ d1 , that
µh1
¯ µ µh1
¯ ¯
2εn−1 y2n ≤ y1 + = + .
d1 δ d1
µ
¯
¯ ¯
Since for i = pn−1 + 1, . . . , pn , si = dn + xn − εxn−1 ≤ dn + 1 yields yi ≥ dn +1 ,
the last dual constraint implies
pn
µhn
¯
y2n ≥ yi − 1 ≥ −1
dn + 1
i=pn−1 +1
2hn εn−1 h1
which, combined with the previously obtained inequality, gives µ
¯ dn +1 − d1
2hn εn−1 h1
−1
δ ≤ 2εn−1 , and, since Corollary 3.2 gives dn +1 − d1 − δ ≥ δ , we have
1 2
µ ≤ εn−1 δ.
¯
Acknowledgments We would like to thank an associate editor and the referees for pointing out
the paper [8] and for helpful comments and corrections. Many thanks to Yinyu Ye for precious sug-
gestions and hints which triggered this work and for informing us about the papers [12, 13]. Research
supported by an NSERC Discovery grant, by a MITACS grant and by the Canada Research Chair
program.
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